Bayesian Shrinkage

Overview

Bayesian Shrinkage is a statistical technique used to improve the reliability of player scores, particularly for players with limited tournament data. It prevents players with only a few exceptional performances from being ranked disproportionately high compared to players with consistent tournament participation.

The Problem: Low Sample Size Variance

When players participate in very few tournaments, their average scores can be misleading:

• A player who plays 2 tournaments and scores very well might appear elite
• A player who plays 20 tournaments with the same average is clearly more proven
• The 2-tournament player might have gotten lucky or faced weaker competition

Example Scenario

Player	Tournaments	Average Score	True Skill?
Player A	2	95.0	Uncertain
Player B	18	87.5	Well-established
Player C	5	82.0	Somewhat reliable

Without shrinkage, Player A would rank above Player B, even though their score is based on far less data.

The Solution: Bayesian Shrinkage

Bayesian shrinkage pulls low-tournament players toward the regional mean, reducing the impact of small sample sizes while preserving the ranking signal for high-tournament players.

What is the Regional Mean?

The regional mean is the average performance score of all players in a specific region (e.g., NAE, EU, NAW) for a given season. It's calculated as:

$$\text{Regional Mean} = \frac{\sum \text{All Player Scores in Region}}{\text{Number of Players}}$$

Why it matters:

• Baseline comparison: Provides a reference point for what's "average" performance in that region
• Regional context: Accounts for differences in skill levels between regions (some regions may be more competitive)
• Stability point: Acts as an anchor that prevents extreme scores from dominating rankings

Example:

• NAE regional mean might be 75.0 points (highly competitive)
• EU regional mean might be 68.5 points (different competitive environment)
• A player with 2 tournaments scoring 95 in NAE gets shrunk toward 75.0
• The same player scoring 95 in EU gets shrunk toward 68.5

The regional mean ensures that a player who plays few tournaments is evaluated relative to their specific competitive environment, not against some arbitrary standard.

Mathematical Formula

$$\text{Shrinkage Weight} = \frac{N}{N + P}$$

Where:

• $N$ = Number of tournaments played by the player
• $P$ = Bayesian prior (default tournaments, typically 6-18 based on season length)

$$\text{Shrunk Score} = (\text{Shrinkage Weight} \times \text{Raw Score}) + ((1 - \text{Shrinkage Weight}) \times \text{Regional Mean})$$

How It Works

Step 1: Calculate Shrinkage Weight

The weight determines how much we trust the player's raw score versus the regional average:

Tournaments (N)	Prior (P=6)	Shrinkage Weight	Interpretation
3	6	0.33	Strong shrinkage toward mean
6	6	0.50	Balanced between raw and mean
12	6	0.67	Moderate trust in raw score
18	6	0.75	Strong trust in raw score
30	6	0.83	Very high trust in raw score

Step 2: Apply Regional Mean

The formula blends the player's raw score with the regional mean:

Shrunk Score = (Weight × Raw Score) + ((1 - Weight) × Regional Mean)

Step 3: Dynamic Prior Calculation

The Bayesian prior $P$ is calculated based on the season's total windows:

$$P = \max(6, \min(18, \text{round}(\text{Total Windows} \times 0.4)))$$

This means:

• Short seasons (fewer windows): Prior = 6
• Long seasons (many windows): Prior = up to 18
• Adaptive to competitive season length

Visual Example

Consider a regional mean of 75.0 points:

Player	Raw Score	Tournaments	Shrinkage Weight	Shrunk Score	Change
Elite A	120.0	25	0.81	111.4	-7.2%
Pro B	95.0	12	0.67	88.4	-7.0%
Casual C	110.0	3	0.33	86.7	-21.2%
Lucky D	130.0	2	0.25	88.8	-31.7%

Notice how:

• Elite A (25 tournaments): Minimal shrinkage, score remains elite
• Lucky D (2 tournaments): Heavy shrinkage, suspiciously high score normalized

Benefits

1. Statistical Reliability

• Reduces variance from small sample sizes
• Produces more stable rankings week-to-week

2. Fairness

• Rewards consistent participation
• Prevents "lucky streak" players from dominating

3. Predictive Accuracy

• Shrunk scores better predict future performance
• Less overfitting to outlier results

4. Regional Normalization

• All players are measured against their region's baseline
• Accounts for regional skill distribution differences

Mathematical Properties

Convergence

As tournament count increases, shrunk score converges to raw score:

$$\lim_{N \to \infty} \text{Shrunk Score} = \text{Raw Score}$$

Bounded Effect

The shrinkage effect is always bounded:

• Maximum shrinkage: When $N = 0$, score equals regional mean
• Minimum shrinkage: When $N \to \infty$, score equals raw score

Preservation of Rankings

For players with the same tournament count, their relative rankings are preserved (both are shrunk proportionally).

Relationship to Log Volume Confidence

Bayesian shrinkage works in tandem with Log Volume Confidence:

1. Shrinkage adjusts the score magnitude based on reliability
2. Confidence further scales the final score based on participation volume

$$\text{Final Score} = \text{Shrunk Score} \times \text{Confidence}$$

This two-step process ensures both statistical reliability (through shrinkage) and participation rewards (through confidence).

Real-World Impact

In practice, Bayesian shrinkage:

• Prevents new players with 1-2 good tournaments from appearing in S-Tier
• Stabilizes tier assignments for players near tier boundaries
• Makes rankings more defensible and statistically sound
• Improves the correlation between tier assignment and true skill level